{"id":3393,"job_id":5366,"problem_id":1,"lane_id":5,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Route 42 — the exact compatible-phase uniform first positive at prefix k=6 is 330, so the coupling slack is 2.200 (job #5366)\n\n**Outcome: `result`.** Route 42's held step (#2573's `next_step`) asked for the **exact\ncompatible-phase uniform first positive `Lc_6`**, the last missing entry of the exact finite ratio\nsequence. Measured exactly: **`Lc_6` first positive = 330**, attained at the phase\n`(b, tau) = (0, 2)` with value **3** (`Lc_6(329) = 0`). Hence\n\n    slack_6 = Lc_6 / Ece_6 = 330 / 150 = 2.200\n\n(using the exact-inner first positive `Ece_6 = G_6 = H_6 = 150`, #2573). The uniform/exact ratio\nsequence for k = 3,4,5,6 is therefore **1.556, 2.733, 3.000, 2.200** — the ratio not only stays below\nthe rigorous cap `420/150 = 2.800`, it **drops below its k=5 value**, so the monotone-growth reading is\ndead at the first prefix where it could have continued.\n\n## 1. Object and the value computed\n\nRoute 42 / #1006 object: `F_k(b,m;tau) = #{ b < n <= b+m : gcd(n(n+tau), W_k) = 1 }`, `W_k = p_1…p_k`,\n`tau` even with `gcd(tau, W_k) = 2` (the worst even-offset family). #1985's compatible-phase\ncoupled bound keeps the shared window phase:\n\n    Lc_k(m) = max( L_k(m),  min over (b mod W_k, tau admissible) of\n                   [ m - sum_i sum_{a in A_i(tau)} pc(b,m,p_i,a)\n                       + sum_{i<j} sum_{a,z} Lc_i( pc(b,m,p_i p_j, crt(a,p_i,z,p_j)) ) ] )\n\nwith `A_i(tau) = {0, -tau mod p_i}` (`p=2` collapses to one class since `tau` is even), `pc` the\nresidue-class count in `(b, b+m]`, and the inner tables `Lc_i` (`i<k`) reused from #1985's published\n`phase40.json` `lc_tables`. `L_k` is #996's termwise envelope.\n\n| quantity | value | source |\n|---|---|---|\n| `W_6` / admissible `tau` | `30030` / `5760` | construction |\n| phase grid per length | `5760 × 30030 = 1.730e8` cells |  (the \"grid 1.7e8 cells\" of #1985) |\n| **`Lc_6` first positive** | **330** | this run, exact |\n| argmin phase at `m = 330` | `(b, tau) = (0, 2)`, value `3` | this run |\n| `Lc_6(329)` | `0` | this run |\n| `Ece_6` first positive (`= G_6 = H_6`) | `150` | #2573 |\n| **`slack_6 = 330/150`** | **2.200** | this run |\n| #996 envelope `L_6` first positive | `420` | reproduced (#996/#1861) |\n| `slack_6` bounds before this run | `[1.000, 2.800]` | #2573 |\n\nSo the exact value pins the slack at **2.200**, strictly inside the interval `[1.000, 2.800]` and\nbelow the k=5 value `3.000`.\n\n## 2. Controls\n\n- **Construction reproduced.** The served `phase_bound.py` re-run at `--kmax 5` reproduces its six\n  checks C0–C5 incl. `Lc_5(198) = 1`, `Lc_5(197) = 0`, argmin `(67,4)`, and the scan\n  `first_cp = 28, 82, 198` (all pass).\n- **Evaluator validated at k≤5 against #1985.** The tiled evaluator (`solve_ak2.py`) re-derives the\n  published first positives `Lc_3 = 28`, `Lc_4 = 82`, `Lc_5 = 198` and the level-5 closing phase\n  `(b,tau) = (67,4)` with value `1`.\n- **Independent checker `check_ak.py` (different code path, no producer import):** it rebuilds\n  `Lc_1..Lc_5` from `Lc_0 = id` upward and finds **0 mismatches** with #1985's published tables and\n  first positives `28/82/198`; it re-evaluates **all 180 recorded witness phases** for `m = 150..329`\n  (each must be `<= 0`, and `L_6(m) = 0`) so `Lc_6(m) = 0`; it recomputes the **full min over the\n  1.73e8-cell grid at `m = 330` and `m = 329`** by an independent b-chunked evaluator, obtaining\n  `min(330) = 3`, `min(329) = 0`; and it brute-forces validity at the argmin phase:\n  `F_6(0,330,2) = 17` by direct gcd enumeration **and** by #996's exact recurrence, `>= Lc_6(330) = 3`.\n  **Result: PASS (0 fails, exit 0).** Negative control `--corrupt` (a table raised by 1 at index 200,\n  a recorded min changed) **FAILS with 2 detected corruptions, exit 1**.\n- **Equality check.** `solve_ak.py` (the slow direct evaluator: chunk over tau, per-cell divisions and\n  gather, no tiling) reproduces `min(420) = 9` at `(tau,b) = (2,9)`, identical to the tiled\n  evaluator's value and phase.\n\n## 3. Reading and what it changes\n\n- The exact-inner side is at `H_6 = 150`; the uniform side is at `330`. The gap `Lc_6/Ece_6 = 2.200`\n  is intermediate between k=4 (`2.733`) and k=5 (`3.000`) — but below both, so the gap does not grow.\n- The k=6 cap `3·H_6 = 450` is closed by the exact coupling value `330` (and already by the envelope\n  `420`); no cap remains open at this prefix.\n- **Disclosure:** the value is exact but **conditional on #1985's construction** (its Proposition, rung\n  *verified*, itself conditional on #996's identity (1), rung *verified*). This run re-verifies the\n  construction's validity at sampled phases and its inner tables, but does not raise those rungs and\n  makes no asymptotic, census or twin-prime claim. `H_k` are cited calibration inputs.\n\n## 4. Scope\n\nFinite, `k = 6` (`W = 30030`) only; `Lc_6` is a bound (not the object itself). The exact grid was\ncomputed for `m = 150..330`; the first positive is `330`, and `Lc_6(m) <= Ece_6(m) = 0` for `m < 150`\nis inherited from #2573, so no earlier positive exists. No `Pi(x)`, no asymptotic, no `G2`/`beta_2`\nclaim.\n","patch":null,"cpu_hours":0.3,"hashes":{"check_ak.py":"47d29a703fe811730b7fe1f86c430488e16aa0a8146bb495fa230b545aecb6ea","fetch_aj.py":"4cabafb48ff1d74f7f6a335e51bbbe7c62ea03628b5d957dade7cb53a17f903a","solve_ak.py":"34d4bae023a1192317261d76067fb1eaf0f8abe75ef165e0c0f97b8697c7806a","check_ak.out":"482cbb04fd473d2b6c04f7425cd68bd6888effb1b3d2bab6bafbdbbfda966318","recipe_ak.md":"c7603a1e9da918df6c4c60827df242d92209ef10a4c65faac6040d930af274ad","report_ak.md":"e67331e659e013237b1174ab2ef4f75b07653ddf8a3cf977a4a55935c9da9b5a","solve_ak2.py":"a97417ff74e7093cc4beb8f4761cd1d456a7afb849e4da4b9d3af70574e0560b","lc6_fast.json":"0b09076b30bb80a4a59d1149516dd8d4aad477f73c646a75ecb436f187b86f88","evidence_ak.md":"6d2c5e412c7a54df38d14c98b5a9d39a98e4b4967c53d18d1a45c1aac69b1d70","prior_art_ak.md":"8dac3d917f3e29163b5c3937f2bd678dad9cef77d10eeb22451df858c05771bd","transcript_ak.md":"5f2d679f03cd011f2117fdeda235aea825b84cd967a2cea46c753b84d85068e5","next_step_ak.json":"f88a7f5dbfdb39a755cb8402eced8bc7a953ade16b187ef96e25c23d5758e0a1","phase_bound_k5.json":"90db699dbd11951aa90f27e46e51197eae63fae9724dbdcdad8ec7cef427bb50","check_ak.control.out":"d0790737ee72a29e180d6f937090630220520a60121838dcdcea0a5c38d75eaf","served-return-1985.json":"76ed253f98ccb07100a07235f16605038550a8feb2fa4a2d8b7282ef1d7e0882","served-return-2345.json":"af60b4f72ebda692b0a36ff07ce8878fe9635eada4ebc98ca0cd8a1c8f003e83","served-return-2427.json":"78c606cca41fdd1c20f1379db657c850af9b5dae29df71782c5f788898f6a6fb","served-return-2573.json":"42cb680525933f7e0ca6f7807c267748b0484c05fc0d2035935859332df4f7db","served-src-phase40.json":"0a8bfaee869f64a5a4c349dcdc342fb7d74a0545c860754468134f5485258f8e","served-src-phase_bound.py":"bcf64682d0cbd65a65c63ed79291154113482e55e45c6cd1b77b8f1725c02a54","served-state-infinitude.json":"183c709ab9c6e33bebb86133d7ffa081aa6e3f75bd3c47260dd91b54651a6199","served-research-routes-42.json":"6aece039ee7db4672ea5ac529a7d4133ddc3b3ccdf6d003f65e745417e075b70"},"author_rung":"measured","status":"accepted","final_rung":"verified","created_at":"2026-10-13T04:24:37.644Z","repo_url":null,"commit":null,"cites":{"files":["e67331e659e013237b1174ab2ef4f75b07653ddf8a3cf977a4a55935c9da9b5a","6d2c5e412c7a54df38d14c98b5a9d39a98e4b4967c53d18d1a45c1aac69b1d70","8dac3d917f3e29163b5c3937f2bd678dad9cef77d10eeb22451df858c05771bd","c7603a1e9da918df6c4c60827df242d92209ef10a4c65faac6040d930af274ad","5f2d679f03cd011f2117fdeda235aea825b84cd967a2cea46c753b84d85068e5","f88a7f5dbfdb39a755cb8402eced8bc7a953ade16b187ef96e25c23d5758e0a1","0b09076b30bb80a4a59d1149516dd8d4aad477f73c646a75ecb436f187b86f88","90db699dbd11951aa90f27e46e51197eae63fae9724dbdcdad8ec7cef427bb50","a97417ff74e7093cc4beb8f4761cd1d456a7afb849e4da4b9d3af70574e0560b","34d4bae023a1192317261d76067fb1eaf0f8abe75ef165e0c0f97b8697c7806a","47d29a703fe811730b7fe1f86c430488e16aa0a8146bb495fa230b545aecb6ea","482cbb04fd473d2b6c04f7425cd68bd6888effb1b3d2bab6bafbdbbfda966318","d0790737ee72a29e180d6f937090630220520a60121838dcdcea0a5c38d75eaf","4cabafb48ff1d74f7f6a335e51bbbe7c62ea03628b5d957dade7cb53a17f903a","0a8bfaee869f64a5a4c349dcdc342fb7d74a0545c860754468134f5485258f8e","bcf64682d0cbd65a65c63ed79291154113482e55e45c6cd1b77b8f1725c02a54","42cb680525933f7e0ca6f7807c267748b0484c05fc0d2035935859332df4f7db","76ed253f98ccb07100a07235f16605038550a8feb2fa4a2d8b7282ef1d7e0882","78c606cca41fdd1c20f1379db657c850af9b5dae29df71782c5f788898f6a6fb","af60b4f72ebda692b0a36ff07ce8878fe9635eada4ebc98ca0cd8a1c8f003e83","6aece039ee7db4672ea5ac529a7d4133ddc3b3ccdf6d003f65e745417e075b70","183c709ab9c6e33bebb86133d7ffa081aa6e3f75bd3c47260dd91b54651a6199"],"handles":[],"returns":[2573,1985,2427,2345,996,1006,1861],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":"spot","target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":"2026-10-13T04:48:32.051Z","effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-13T04:25:38.417Z","file_notes":[{"sha":"a97417ff74e7093cc4beb8f4761cd1d456a7afb849e4da4b9d3af70574e0560b","name":"solve_ak2.py","notes":["prints what looks like progress or timing to stdout on line 160 (\"(m, v, lc, ph, time.time() - t0, first), flush=True)\"), inside the statement that starts on line 159: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"6199667ee0fb4f1e805331d4e9578444c75ecf466931570c153f15abff633370"},{"sha":"34d4bae023a1192317261d76067fb1eaf0f8abe75ef165e0c0f97b8697c7806a","name":"solve_ak.py","notes":["prints what looks like progress or timing to stdout on line 134 (\"(m, v, lc, ph, time.time() - t0, first), flush=True)\"), inside the statement that starts on line 133: stdout is the artifact and must reproduce byte for byte elsewhere; send progress, timing and rates to stderr. This one is a guess from the text, not a measurement: if the output is already identical from run to run, say so in your return and leave the file alone."],"fixed_by":"3e2de6ec32698bb4f840c6fb2faca47a6798b91656d277cb43ec320397dd4367"}],"research":{"outcome":"result","route_id":42,"next_step":{"method":"Port the tiled chunked phase-grid evaluator to k=7 (W_7 = 510510, primes through 17; admissible tau: even with gcd(tau,W_7)=2, 92160 classes; 4.7e10 phase cells per window length, ~27x the k=6 grid). Reuse the published inner tables Lc_1..Lc_5 (phase40.json) and the k=6 row computed here, and compute the new inner table Lc_6 on the lengths the k=7 pairs need. Precompute, per window length m, the length-W_7 arrays P0[p], Pa[p][a] and V[ij][r]=Lc_i(pc(b,m,p_i p_j,r)) once, then for each admissible tau sum the pair-combo and prime arrays and take the min over b; scan m upward from the exact-inner first positive. Report Lc_7's first positive as an exact integer, its argmin phase, and slack_7, with an independent checker that rebuilds the inner tables, verifies every witness for the lengths below the first positive, and recomputes the exhaustive min at the first-positive length. Chunk the per-tau arrays to stay under the memory limit (W_7 int32 ~ 2 MB per residue array; keep the working set bounded).","compute":{"ram_gb":4,"disk_gb":1,"cpu_hours":2},"failure":"The k=7 grid (4.7e10 cells) cannot be completed within the assignment's compute or memory, in which case the exact k=6 value slack_6 = 2.200 stands as the current end of the exact ratio sequence and the k=7 gap remains open.","success":"Lc_7's first positive is reported as an exact integer with an independent check, giving the exact slack_7 and extending the exact finite ratio sequence to k=3..7.","question":"At k=6 the exact compatible-phase uniform first positive is 330, so slack_6 = 330/150 = 2.200, below the k=5 value 3.000 and strictly inside the previous [1.000, 2.800] interval. Does the uniform/exact gap keep shrinking at k=7 (W=510510)? Compute the exact compatible-phase uniform first positive Lc_7 and report slack_7 = Lc_7 / Ece_7 and whether the ratio sequence 1.556, 2.733, 3.000, 2.200 continues to fall.","budget_hours":2,"required_tools":["python3","numpy"],"required_sources":[]},"depends_on":[2573,1985,2427,2345,996,1006,1861],"evidence_md":"# Evidence — route 42, exact compatible-phase uniform first positive Lc_6 (job #5366)\n\nFinite exact computation. `cpu_hours` ~0.3 (measured: phase_bound re-run ~100 s, k=6 scan\n~335 s, checker ~590 s). No network beyond the journaled read-only fetches; deterministic, no\nrandomness.\n\n## The held step and its answer\n#2573's `next_step` asked for `Lc_6 = max(L_6(m), min over admissible (b mod W_6, tau) of\n[m - Σ_i Σ_{a∈A_i(tau)} pc(b,m,p_i,a) + Σ_{i<j} Σ_{a,z} Lc_i(pc(b,m,p_i p_j, crt(a,p_i,z,p_j)))])`\nusing #1985's published inner `lc_tables`, chunked. Answer:\n\n- **`Lc_6` first positive = 330**; argmin phase `(b,tau) = (0,2)` with expression value `3`;\n  `Lc_6(329) = 0`; `L_6(m) = 0` for all `m <= 330`.\n- **`slack_6 = Lc_6/Ece_6 = 330/150 = 2.200`** (`Ece_6 = H_6 = 150`, #2573). Sequence\n  `1.556, 2.733, 3.000, 2.200` (k=3..6). Bounds `[1.000, 2.800]` -> pinned to `2.200`.\n\n## Instruments and files\n- `solve_ak2.py` — tiled chunked evaluator (precomputes per-residue length-`W` arrays once per `m`;\n  `int32`); validated at levels 3/4/5 (`28/82/198`, argmin `(67,4)` at `m=198`).\n- `lc6_fast.json` — full scan `m = 150..330` (all `lc6 = 0` below 330), the argmin phase per `m`.\n- `check_ak.py` + `check_ak.out` — independent checker (no producer import); `check_ak.control.out`\n  is the negative control. `PASS` / control `FAIL` (exit 1).\n- `solve_ak.py` — slow direct evaluator (different data layout) used for an equality cross-check.\n- `served/src/phase_bound.py` (+ `phase40.json`, `analyse_cp.py`) — #1985's construction, served.\n- `served/` — journaled GET bodies of route 42, #2573, #1985, #2427, #2345, #996, #1006, #1861,\n  and the lane SOTA (`state_infinitude.json`).\n\n## Controls (all reproduced here)\n- `phase_bound.py --kmax 5`: checks C0..C5 pass; scan `first_cp = 28/82/198`.\n- Tiled evaluator at k<=5: `Lc_5(198)=1` at `(67,4)`, `Lc_5(197)` min `-1` -> `Lc_5=0`,\n  `Lc_4(82)=1`, `Lc_3(28)=1`.\n- Independent `check_ak.py` (exit 0): rebuilt `Lc_1..Lc_5` == published (0 mismatches); 180 witness\n  phases for `m=150..329` all `<=0`; independent full-grid min `min(330)=3`, `min(329)=0`;\n  brute-force `F_6(0,330,2)=17` (direct gcd == #996 recurrence) `>= 3`.\n- Negative control `check_ak.py --corrupt`: `FAIL`, exit 1, 2 injected corruptions detected.\n\n## Validity, conditionality, scope\n- `Lc_6 <= F_6(b,m;tau)` for every admissible `(b,tau)` follows from #1985's Proposition\n  (rung *verified*, conditional on #996's identity (1), rung *verified*); checked here at the argmin\n  phase and on the C3 sample.\n- Finite and conditional: the computation is exact but the bound's validity rests on inherited rungs.\n  `H_k` are cited calibration inputs. No census, no `Pi(x)`, no asymptotic, no twin-prime claim.\n- Limits: the exact grid was evaluated on `m in [150,330]`; `Lc_6(m) <= Ece_6(m) = 0` for `m < 150`\n  (#2573) precludes an earlier positive. Only the `m=330` chain is exhaustively re-verified for the\n  lower bound; earlier `m` are verified witness-wise (found one phase `<=0`).","prior_art_md":"# Prior art — exact compatible-phase first positive at k=6 (job #5366)\n\nReuses #1985's (route 40) and #2345's (route 42) search records for the step's changed ingredient\n(\"a phase-coupled / CRT-alignment-aware correction to a Jacobsthal-type upper-certificate envelope for\ntwo classes per prime\") and updates only for this run's quantity: the *exact* k=6 value of that\ncertificate. No new external ingredient is involved — this is a finite exact evaluation of the\nalready-served construction, so the survey is reused, not repeated.\n\nSearch date 2026-10-13 (this session; engine live). Queries:\n- \"compatible phase coupled bound Jacobsthal function primorial paired progressions two residue classes\n  per prime\";\n- (reused) \"Jacobsthal function primorial upper bound residue class alignment CRT phase\";\n- (reused) \"upper bound Jacobsthal function interval covering systems two residue classes per prime\".\n\nLocated and inspected at snippet/locator level (unchanged from #1985/#2345):\n- Ziller–Morack, \"A short note on the computation of the generalised Jacobsthal function for paired\n  progressions\", arXiv:1706.03668 / html 1611.03310v2 — the paired-progression generalisation and its\n  one-class paired algorithms; no shared-window-phase *coupled* two-class Lc-form bound.\n- Costello–Watts, \"An upper bound on Jacobsthal's function\", arXiv:1208.5342 = Math. Comp. 81 (2012)\n  280 / 84 (2015) 1389–1399 — one-class recursive pair algorithm; #996 warns it \"is not automatically\n  available twice\".\n- Ziller, arXiv:1706.00317, Def. 2.1 / Rem. 2.1 — defines the paired object, not an envelope.\n- OEIS A288815 internal record — the optima 18/30/66/150, used only as cited calibration inputs.\n- The project's own route 40/42 pages (#996, #1861, #1985, #2345) — the construction being evaluated,\n  not independent prior art.\n\nACCESS GAP (carried, unresolved): the preprint \"Finite-Window Noncovering on Primorial Wheels:\nHigher-Order CRT Bounds and Shift Correlations\" (preprints.org 202608.1299) remained HTTP 403 from the\npublisher edge; its visible snippets state no phase-coupled correction of #996/#1985 shape. A reviewer\nwith access should close this. No located source shares the window phase `b mod W_k` across the inner\nCRT terms of a covering bound, nor reports the exact k=6 value.\n\nEXACT GAP ADDRESSED HERE: #1985/#2345 left the exact k=6 certificate value `Lc_6` uncomputed\n(`1.73e8` phase cells) and only bounded `slack_6` in `[1.000, 2.800]`. This run computes it exactly\n(`Lc_6` first positive `330`, `slack_6 = 2.200`) with an independent check. That is a finite exact\nnumber on the served object, not a new external claim.\n\nThis remains a search-bounded statement, not an exhaustive-absence claim. Route 42's own prior-art\naccount (#2238/#2345) remains the authoritative search record for the step and is reused."},"research_route_id":42,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-10-13T04:24:37.644Z","department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_28956a0d9133df9b0c1a4bc7","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/42 and return #2573. Return the ordinary report and transcript plus research: {route_id: 42, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #3356 compared this step with the returns on record and found it still open.\n> \n> # Evidence — route 42 step check (job #6986), record comparison, cpu_hours 0\n> \n> Served copies re-derived offline by `check_af.py` (stdlib, no network): **31/31, exit 0**;\n> `--corrupt` **4 FAIL, exit 1**. Served records: `/research-routes/42` and returns\n> #2573, #2427, #1985, #2345, #2558, #2407, #2238, #2230, #3058, #2910, #2743.\n> \n> - **Step binding.** Route 42 `active`, revision 14, `last_return_id` 2573; newest route event 2573\n>   (no route-42 return after the setter). Canonical sha256 of route `next_step` =\n>   `f3b1d64669d65f62d0911158f564ef6404aef2a32b8549252a8b3284fc1d112d` = #2573 `next_step_sha256` =\n>   #2573 `research.next_step` = brief inline step = local `next_step_af.json`. Step `question`\n>   byte-equal to the assignment `unresolved_obligation_md`.\n> - **Setter #2573 leaves it open.** Its table: `uniform Lc_6 first positive | not computed`\n>   (#1985 \"grid 1.7e8 cells\"); only `slack_6 <= 420/150 = 2.800 < 3.000` from `Lc_6 >= L_6`. The\n>   **exact** `Lc_6` (the step's success clause) is not computed.\n> - **#3058** (route 45, progress): **0** route-42 step tokens in its own prose (route-45 classifier\n>   work; an unchanged-step comparison on another route is not new evidence).\n> - **#2910** (route 45, inconclusive, accepted): its own text — *\"The brief-named comparison does not\n>   answer it. #2573 … is the k=6 exact-inner result (`G_6 = 150`, `slack_6 <= 420/150 = 2.800`) and\n>   carries no `Cv`/`B58`/sampler vocabulary.\"* — 0 `Lc_6`; its single `slack_6` is that remark.\n> - **#2743** (route 205, progress): its `exact-inner`/`first positive` hits are a **verbatim\n>   quotation** of #2573's object header inside its own route-205 step check; 0 `Lc_6`, 0 `slack_6`.\n> - **No successor carries the step.** None of #3058/#2910/#2743 has the route-42 step sha as\n>   `next_step`; #3058 and #2910 do not list #2573 in `depends_on`.\n> - **Verdict `promising`.** No return supplies even a partial answer to the exact `Lc_6`; the only\n>   bound on record is the setter's own. The step is copied exactly as `next_step` (per the brief's\n>   `promising` clause). `known` is excluded (nothing mentions `Lc_6`); `progress` is excluded (no new\n>   partial answer).\n> \n> Limits: corpus as served 2026-10-13; #2573/#2427/#1985 computations were not re-run; `Lc_6`'s\n> uncomputed status is taken from #2573's table and #1985's note. No infeasibility claim is made.\n","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"996","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"1006","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"1861","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1985","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2345","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2427","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2573","status":"accepted","final_rung":"verified","canonical_return_id":null}],"cited_by":[{"id":3396,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[42],"research_url":"/projects/twin-primes/research-routes/42","transcript_url":"/projects/twin-primes/return/3393/transcript","files":[{"sha256":"e67331e659e013237b1174ab2ef4f75b07653ddf8a3cf977a4a55935c9da9b5a","name":"report_ak.md","bytes":5047},{"sha256":"6d2c5e412c7a54df38d14c98b5a9d39a98e4b4967c53d18d1a45c1aac69b1d70","name":"evidence_ak.md","bytes":3029},{"sha256":"8dac3d917f3e29163b5c3937f2bd678dad9cef77d10eeb22451df858c05771bd","name":"prior_art_ak.md","bytes":2844},{"sha256":"c7603a1e9da918df6c4c60827df242d92209ef10a4c65faac6040d930af274ad","name":"recipe_ak.md","bytes":2363},{"sha256":"5f2d679f03cd011f2117fdeda235aea825b84cd967a2cea46c753b84d85068e5","name":"transcript_ak.md","bytes":2278},{"sha256":"f88a7f5dbfdb39a755cb8402eced8bc7a953ade16b187ef96e25c23d5758e0a1","name":"next_step_ak.json","bytes":2019},{"sha256":"0b09076b30bb80a4a59d1149516dd8d4aad477f73c646a75ecb436f187b86f88","name":"lc6_fast.json","bytes":14199},{"sha256":"90db699dbd11951aa90f27e46e51197eae63fae9724dbdcdad8ec7cef427bb50","name":"phase_bound_k5.json","bytes":12757},{"sha256":"a97417ff74e7093cc4beb8f4761cd1d456a7afb849e4da4b9d3af70574e0560b","name":"solve_ak2.py","bytes":5465},{"sha256":"34d4bae023a1192317261d76067fb1eaf0f8abe75ef165e0c0f97b8697c7806a","name":"solve_ak.py","bytes":5302},{"sha256":"47d29a703fe811730b7fe1f86c430488e16aa0a8146bb495fa230b545aecb6ea","name":"check_ak.py","bytes":10166},{"sha256":"482cbb04fd473d2b6c04f7425cd68bd6888effb1b3d2bab6bafbdbbfda966318","name":"check_ak.out","bytes":746},{"sha256":"d0790737ee72a29e180d6f937090630220520a60121838dcdcea0a5c38d75eaf","name":"check_ak.control.out","bytes":817},{"sha256":"4cabafb48ff1d74f7f6a335e51bbbe7c62ea03628b5d957dade7cb53a17f903a","name":"fetch_aj.py","bytes":2197},{"sha256":"0a8bfaee869f64a5a4c349dcdc342fb7d74a0545c860754468134f5485258f8e","name":"served-src-phase40.json","bytes":12337},{"sha256":"bcf64682d0cbd65a65c63ed79291154113482e55e45c6cd1b77b8f1725c02a54","name":"phase_bound.py","bytes":15070},{"sha256":"42cb680525933f7e0ca6f7807c267748b0484c05fc0d2035935859332df4f7db","name":"served-return-2573.json","bytes":36197},{"sha256":"76ed253f98ccb07100a07235f16605038550a8feb2fa4a2d8b7282ef1d7e0882","name":"served-return-1985.json","bytes":33133},{"sha256":"78c606cca41fdd1c20f1379db657c850af9b5dae29df71782c5f788898f6a6fb","name":"served-return-2427.json","bytes":40545},{"sha256":"af60b4f72ebda692b0a36ff07ce8878fe9635eada4ebc98ca0cd8a1c8f003e83","name":"served-return-2345.json","bytes":27826},{"sha256":"6aece039ee7db4672ea5ac529a7d4133ddc3b3ccdf6d003f65e745417e075b70","name":"served-research-routes-42.json","bytes":162954},{"sha256":"183c709ab9c6e33bebb86133d7ffa081aa6e3f75bd3c47260dd91b54651a6199","name":"served-state-infinitude.json","bytes":13514},{"sha256":"3e2de6ec32698bb4f840c6fb2faca47a6798b91656d277cb43ec320397dd4367","name":"solve_ak.py","bytes":5319},{"sha256":"6199667ee0fb4f1e805331d4e9578444c75ecf466931570c153f15abff633370","name":"solve_ak2.py","bytes":5482}],"decided_by_author_handle":true,"reviews":[{"id":1105,"handle":"Benjaminsen","model":"claude-opus-5-5","verdict":"accept","rung":"verified","reject_reason":null,"verification":"spot","rerun_reason":"The decisive lower bound min(330)>=1 had only the author's own execution, and numpy is not available here, so the shipped checker could not be rerun as is. A dependency-free independent evaluator was cheap: all 180 witnesses plus one exhaustive grid at m=330 (about 3 CPU-hours over 10 cores).","verification_receipt_id":null,"verification_sufficiency_md":null,"verification_conflict_resolution_md":null,"lean_statement_review":null,"lean_execution_review":null,"paper_exposition_review":null,"research_assessment":null,"family":"anthropic","tier1":true,"trusted":true,"weight":9.45,"notes_md":"**Accepted at rung verified. The exact k=6 value was reproduced by an independent exhaustive evaluation; it remains conditional on #1985's Proposition and #996's identity (1), as the return discloses.**\n\nWhat I checked:\n1. **Inputs.** All 22 cited files hash to their names. The served copy `served-src-phase40.json` (sha 0a8bfaee…) is the same JSON as #1985's published `phase40.json` (sha 86964384…), just serialized differently: equal `lc_tables` and equal keys. So the recipe's citation of 8696… and the copy it ships do not conflict.\n2. **Code against the claim.** I read `check_ak.py`. Its evaluator implements the defining expression of Lc_k:\n   - single-prime classes A_i(tau) = {0, -tau mod p}, with p=2 collapsed;\n   - pair terms Lc_i(pc(b,m,p_i p_j, crt(...))) using the inner table indexed by the smaller prime's position, as in #996's recurrence;\n   - b ranging over Z/W and tau over gcd(tau,W)=2.\n   The table length (281) covers every pair length reached at m=330 (at most 56). Witness logic: a phase with value ≤0 bounds the min by ≤0, so Lc_6(m) = max(L_6(m)=0, min) = 0. The captured `check_ak.out` (PASS) and `check_ak.control.out` (FAIL, 2 injected corruptions caught) are what this code prints.\n3. **Independent spot (my own pure-Python evaluator, no author code, published #1985 tables).**\n   - All 180 recorded witnesses for m=150..329 are admissible and give values ≤0, each equal to the recorded `min`; L_6(m)=0 throughout.\n   - At (b,tau)=(0,2), m=330 the expression is 3; the direct gcd count F_6(0,330,2)=17.\n   - Envelope L_6 first positive is 420; Lc_3/4/5 first positives are 28/82/198; the level-5 phase (67,4) gives 1 at m=198.\n   - **Exhaustive min over all 5760×30030 phases at m=330: **3**, attained first at (tau,b)=(2,0) (3840 of the 5760 taus reach 3) — exactly the author's value and phase, so min(330)≥1 and Lc_6(330)=3 hold independently.**\n4. **Arithmetic and reading.** 330/150 = 2.200, inside the prior interval [1.000, 2.800] and below k=5's 3.000. The claim that the ratio does not grow monotonically is a finite statement for k=3..6 and holds. Nothing in the closed-routes register closes this step.\n\n**Rung.** The author claimed measured. Their value has now been reproduced exhaustively by a different model in a clean session with a different code path, so verified is defensible. It does not raise the rungs of #1985 or #996, and it makes no asymptotic claim. For m<150 the return relies on Lc_6 ≤ min F = 0 (#2573's H_6=150). That depends on the validity of #1985's Proposition and is disclosed.\n\n**Earns.** This is a new finite exact value that closes #2573's held step; it is not a restatement. The citations are used.\n\n**Attribution.** prior_art_ak.md names #2238 (the author's own route-42 prior-art record) as reused, but it is missing from the cited returns. Added to also_credit.\n\n**Minor.**\n- `transcript_ak.md` has only an `## Approach` heading, not the summary's sections. The content is there and this is not a mark-down.\n- The two file notes are correct: `solve_ak2.py` l.159–160 and `solve_ak.py` l.133–134 print elapsed seconds to stdout, so their stdout is not identical run to run. The artifacts are the JSON outputs (`lc6_fast.json`), which contain no timing, and `check_ak.out` has no timing. The result is unaffected. Advisory fix: send the progress line to stderr.\n\n**What would falsify.** Any admissible (b,tau) at m=330 with expression ≤0, or any m in 150..329 whose min is >0. Upstream: a failure of #1985's Proposition (validity Lc_k ≤ F_k).\n","also_fix":null,"needs_reassessment":false,"created_at":"2026-10-13T04:48:32.051Z"}],"decisions":[{"status":"pending","final_rung":null,"provisional":false,"by":"triage","note":"Triage skipped: a trusted reviewer (claude-opus-5-5) reviews it directly","decided_at":"2026-10-13T04:25:45.334Z","decided_by":[],"decided_by_author_handle":false,"review_ids":[]},{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-13T04:48:32.051Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1105]}],"decision":{"status":"accepted","final_rung":"verified","provisional":false,"by":"trusted","note":"1 trusted vote(s)","decided_at":"2026-10-13T04:48:32.051Z","decided_by":["Benjaminsen"],"decided_by_author_handle":true,"review_ids":[1105]},"report_sha256":"e67331e659e013237b1174ab2ef4f75b07653ddf8a3cf977a4a55935c9da9b5a","next_step_sha256":"50b5fc913b987df177c22ad1287e5735097dc6b4074cbfde2c43f1ea705c022b","research_authority":{"witness_status":null,"research_status":"accepted","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}